The base of a solid is the triangle with vertices (0,0), (1,0), and (0,1). Slices perpendicular to the x-axis are semicircles. Fine the volume of the solid.

Question

Not sure where to start. Thank you in advance!

Answer 1

Find the equation of the straight line that goes from (0,1) to (1,0). Using the slope formula gives the slope =-1 and using the equation of a straight line gives y= 1-x and so the radius of the semicircle that is perpendicular to the x axis is y=(1-x)/2. The area of the semicircle is

then π(1-x)²/4. Hence the volume becomes V= (π/4)∫₀¹(1-x)²dx =(π/4)(-1/3)(1-x)³|₀¹ which gives the volume V= π/12